= Solution
In the stated gravity-dominated approximation, neglect the pressure force in the radial momentum equation. A steady outward <stellar wind> then obeys
$$
v\frac{dv}{dr}=-\frac{GM_*}{r^2},\qquad
\frac{v^2}{2}-\frac{GM_*}{r}=C.
$$
At $R_e=R_{\rm esc}$, the <escape velocity> is $v_e=\sqrt{2GM_*/R_e}$, so $C=0$. The <ballistic marginal-escape stellar wind> has
$$
\boxed{v(r)=\sqrt{\frac{2GM_*}{r}}
=v_e\left(\frac r{R_e}\right)^{-1/2}.}
$$
The outward flow decelerates and approaches zero speed at infinity; it is exactly at the escape threshold in this pressure-negligible model. An initial speed above that threshold gives a nonzero asymptotic speed.
Adiabaticity alone would instead conserve the full <Bernoulli equation> $v^2/2+h-GM_*/r=B$. If finite pressure is retained and $v_e$ equals the mechanical escape speed, then $B=h_e$, not zero. With $p=K\rho^\gamma$ and <conservation of mass>, the more general implicit solution is
$$
\frac{v^2}{2}-\frac{GM_*}{r}
+h_e\left(\frac{R_e^2v_e}{r^2v}\right)^{\gamma-1}=h_e.
$$
It needs $h_e$ and $\gamma$ in addition to the escape-speed datum. The boxed explicit power law is the leading gravity-dominated result, not an exact solution for an arbitrary warm adiabatic wind.
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